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The average rank of elliptic $n$-folds

Published 1 Oct 2010 in math.NT and math.AG | (1010.0152v2)

Abstract: Let $V/\mathbb{F}_q$ be a variety of dimension at least two. We show that the density of elliptic curves $E/\mathbb{F}_q(V)$ with positive rank is zero if $V$ has dimension at least 3 and is at most $1-\zeta_V(3){-1}$ if $V$ is a surface.

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